Abstract

We study mathematical properties of internal three-dimensional flows of incompressible heat-conducting fluids with stick-slip boundary conditions, which state that the fluid adheres to the boundary until a certain criterion activates the slipping regime on the boundary. We look at this type of activated boundary condition as at an implicit constitutive equation on the boundary and establish the long-time and large-data existence of weak solutions for the incompressible three-dimensional Navier–Stokes–Fourier system with the viscosity and the heat conductivity depending on the temperature (internal energy). It is essential for our approach to know that the pressure, i.e., the quantity that is a consequence of the fact that the material is incompressible, is globally integrable. While this requirement is in the case of unsteady flows subject to a no-slip boundary condition open for most incompressible fluids, we show that this difficulty can be successfully overcome if one replaces the no-slip boundary condition by a stick-slip boundary condition. The result relies also on the approach developed in Bulíček et al. (Nonlinear Anal. Real World Appl. 10, 992–1015, 10).

Mathematics Subject Classification (2010)

Notes

Acknowledgments

The both authors acknowledge the support of the ERC-CZ project LL1202 financed by the Ministry of Education, Youth and Sports of the Czech Republic.

The authors acknowledge the membership to the Nečas Center for Mathematical Modeling (NCMM) and to the Charles University Center for Mathematical Modeling, Applied Analysis and Computational Mathematics (MathMAC).